Optimal. Leaf size=45 \[ \frac{b^2-4 a c}{40 c^2 d^6 (b+2 c x)^5}-\frac{1}{24 c^2 d^6 (b+2 c x)^3} \]
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Rubi [A] time = 0.0328772, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {683} \[ \frac{b^2-4 a c}{40 c^2 d^6 (b+2 c x)^5}-\frac{1}{24 c^2 d^6 (b+2 c x)^3} \]
Antiderivative was successfully verified.
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Rule 683
Rubi steps
\begin{align*} \int \frac{a+b x+c x^2}{(b d+2 c d x)^6} \, dx &=\int \left (\frac{-b^2+4 a c}{4 c d^6 (b+2 c x)^6}+\frac{1}{4 c d^6 (b+2 c x)^4}\right ) \, dx\\ &=\frac{b^2-4 a c}{40 c^2 d^6 (b+2 c x)^5}-\frac{1}{24 c^2 d^6 (b+2 c x)^3}\\ \end{align*}
Mathematica [A] time = 0.0167315, size = 43, normalized size = 0.96 \[ \frac{\frac{b^2-4 a c}{40 c^2 (b+2 c x)^5}-\frac{1}{24 c^2 (b+2 c x)^3}}{d^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 42, normalized size = 0.9 \begin{align*}{\frac{1}{{d}^{6}} \left ( -{\frac{1}{24\,{c}^{2} \left ( 2\,cx+b \right ) ^{3}}}-{\frac{4\,ac-{b}^{2}}{40\,{c}^{2} \left ( 2\,cx+b \right ) ^{5}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.45589, size = 134, normalized size = 2.98 \begin{align*} -\frac{10 \, c^{2} x^{2} + 10 \, b c x + b^{2} + 6 \, a c}{60 \,{\left (32 \, c^{7} d^{6} x^{5} + 80 \, b c^{6} d^{6} x^{4} + 80 \, b^{2} c^{5} d^{6} x^{3} + 40 \, b^{3} c^{4} d^{6} x^{2} + 10 \, b^{4} c^{3} d^{6} x + b^{5} c^{2} d^{6}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.47657, size = 211, normalized size = 4.69 \begin{align*} -\frac{10 \, c^{2} x^{2} + 10 \, b c x + b^{2} + 6 \, a c}{60 \,{\left (32 \, c^{7} d^{6} x^{5} + 80 \, b c^{6} d^{6} x^{4} + 80 \, b^{2} c^{5} d^{6} x^{3} + 40 \, b^{3} c^{4} d^{6} x^{2} + 10 \, b^{4} c^{3} d^{6} x + b^{5} c^{2} d^{6}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.14445, size = 105, normalized size = 2.33 \begin{align*} - \frac{6 a c + b^{2} + 10 b c x + 10 c^{2} x^{2}}{60 b^{5} c^{2} d^{6} + 600 b^{4} c^{3} d^{6} x + 2400 b^{3} c^{4} d^{6} x^{2} + 4800 b^{2} c^{5} d^{6} x^{3} + 4800 b c^{6} d^{6} x^{4} + 1920 c^{7} d^{6} x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1946, size = 50, normalized size = 1.11 \begin{align*} -\frac{10 \, c^{2} x^{2} + 10 \, b c x + b^{2} + 6 \, a c}{60 \,{\left (2 \, c x + b\right )}^{5} c^{2} d^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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